Hello rakan, None of the check boxes on that screen are system related. the "core software" is the actual os and can only be removed from the berry using javaloader. so your safe to remove the ones you have checked.

With the BBH tool you are using it will not show up. Since it cant be removed with BBH every check box can be checked or removed and the berry will still boot and load. It just wont have many apps, only the default core apps like Phone, address book, and browser.